Motion of Solid and Fluid Bodies. 157 
the triple integrals giving the equations of equilibrium, and the double integrals 
giving the conditions at the limits. If the density be expressed by «¢, so that 
dm = edadydz, we have 
a = — «Y= +, — €Z = = (9) 
which are the well-known equations of hydrostatics. It appears from these equa- 
tions that if the forces (x, y, z) be zero at all points of the fluid, the quantity p 
must be constant ; and vice versa, if p be constant for all points of the fluid, the 
forces x, y, z must be zero; in such a case the function v, will not give any inde- 
finite equations of equilibrium, but only the particular condition at the limits ex- 
pressed by the double integrals, which is the same as that found by Lagrange, 
and expresses that there must be a constant normal pressure at each point of the 
limiting surface, equal to p, in order that the fluid should remain in equilibrium. 
The hydrodynamical equations corresponding to (9) are 
hes Fai A 
dp ay 
Te = ex + = 
°F etaate Fh ot B (10) 
which are the equations commonly used in hydrodynamics, and, together with the 
equations (9), are usually deduced from the principle of equality of pressure, of 
which principle they are merely the mathematical expression: but the fact that 
this principle is often not true in the case of fluids in motion, would lead us to 
suppose that though the equations may do well enough as a first approximation, 
yet that in hydrodynamics they are insufficient ; in fact we have, in general, 
v=v,+v,+Vv.+ &e. 
and the terms v,, v,, &c. give rise, in the equations of equilibrium and motion, to 
differential coefficients of the first, second, &c., order ; and in case the supposition 
v =v, becomes insufficient, we must then make v = v,-+ v, and add to the 
differential equations of equilibrium and motion, arising from v,, the additional 
terms produced by v,. I shall return to this subject in a subsequent part of this 
paper, when I have determined the form of the terms produced by v,. 
It appears from (4) that 
v= (00 8, (¢')?-p’sinddpdode, 
